Element-wise vector inversion

• October 31st 2009, 05:04 AM
fp04
Element-wise vector inversion
Hello,

I have a problem of needing to perform an operation on a vector which results in each element being inverted. i.e.

f(x) = y, where

x = [x1 x2 ... xN] and y = [1/x1 1/x2 ... 1/xN]

I'd really appreciate all of your help!

Fran
• October 31st 2009, 05:08 AM
tonio
Quote:

Originally Posted by fp04
Hello,

I have a problem of needing to perform an operation on a vector which results in each element being inverted. i.e.

f(x) = y, where

x = [x1 x2 ... xN] and y = [1/x1 1/x2 ... 1/xN]

I'd really appreciate all of your help!

Fran

I don't get it: what is the problem? Of course, you can apply your function f ONLY on vectors whose entries are all $\neq 0$ , but for that I can't see any problem

Tonio
• October 31st 2009, 05:32 AM
fp04
Sorry, maybe I wasn't clear. Let me give you the entire problem I'm solving....

I work in the field of MRI Physics. I need to perform the following optimisation:

min{m1,m2} l2norm(Fm1 - y1) + l2norm(Fm2 - y2) + lambda* l1norm(f(m1,m2))

F is the Fourier matrix, m1 and m2 are vectors of length of 512^2, y1 and y2 are vectors of the same length and lambda is a normalisation constant.

The function f divides vectors m1 and m2 in an elementwise manner.

As the vectors are so large I would like to use a conjugate gradient method to perform the optimisation. Therefore, I need to take the derivative of the function w.r.t m1 and m2. However, the elementwise division of m1 and m2 is undifferentiable.

So, I need to rewrite the elementwise division of two vectors in a way which is differentiable. Therein lies the problem!

Is this clear...?
• October 31st 2009, 06:16 AM
tonio
Quote:

Originally Posted by fp04
Sorry, maybe I wasn't clear. Let me give you the entire problem I'm solving....

I work in the field of MRI Physics. I need to perform the following optimisation:

min{m1,m2} l2norm(Fm1 - y1) + l2norm(Fm2 - y2) + lambda* l1norm(f(m1,m2))

F is the Fourier matrix, m1 and m2 are vectors of length of 512^2, y1 and y2 are vectors of the same length and lambda is a normalisation constant.

The function f divides vectors m1 and m2 in an elementwise manner.

As the vectors are so large I would like to use a conjugate gradient method to perform the optimisation. Therefore, I need to take the derivative of the function w.r.t m1 and m2. However, the elementwise division of m1 and m2 is undifferentiable.

So, I need to rewrite the elementwise division of two vectors in a way which is differentiable. Therein lies the problem!

Is this clear...?

Not to me but perhaps someone else knows this stuff.

Tonio